1. **State the problem:** You have the expression $4^{\log_2(2x)}$ and want to simplify or evaluate it. 2. **Recall the formula and rules:** - The expression involves an exponent with a logarithm. - Use the property of logarithms and exponents: $a^{\log_b(c)} = c^{\log_b(a)}$ or rewrite bases to a common base. - Note that $4 = 2^2$, so rewrite the base 4 as $2^2$. 3. **Rewrite the expression:** $$4^{\log_2(2x)} = (2^2)^{\log_2(2x)}$$ 4. **Use the power of a power rule:** $$= 2^{2 \cdot \log_2(2x)}$$ 5. **Use the logarithm power rule:** $$= 2^{\log_2((2x)^2)}$$ 6. **Simplify the inside of the logarithm:** $$(2x)^2 = 4x^2$$ 7. **Apply the property $2^{\log_2(y)} = y$:** $$= 4x^2$$ **Final answer:** $$4^{\log_2(2x)} = 4x^2$$